The second probability is easy to analyse, so we will concentrate on bounding the first one. Similarly to Lemmas F.1 and F.3 from the supplementary material of [1]} one can show that
\(\mathbb {P}_{\theta _0}\bigg ( \sup _{u \in \mathcal {C}(s,3+4/\gamma )} \sup _{\xi \in \Theta }\frac{|H_{u}(\theta _0)-H_{u}(\xi )|}{\Vert  u \Vert _2^2} \ge \frac{l_{\text{min}}}{4}\bigg )\le \\\mathbb {P}_{\theta _0}\bigg ( 3(c_0+2)^2\sup _{u \in \mathcal {K}(2s)} \sup _{\xi \in \Theta }\frac{|H_{u}(\theta _0)-H_{u}(\xi )|}{\Vert  u \Vert _2^2} \ge \frac{l_{\text{min}}}{4}\bigg )\le \\21^{2s} \left( p^{2s} \wedge \left( \frac{ep}{2s} \right)^{2s} \right) \sup _{u \in \Theta } \mathbb {P}_{\theta _0}\bigg ( \sup _{\xi \in \Theta }\frac{|H_{u}(\theta _0)-H_{u}(\xi )|}{\Vert  u \Vert _2^2} \ge \frac{l_{\text{min}}}{36(c_0+2)^2}\bigg )\) 
